I generally define words by the dictionary as that's what we've xollextively agreed to go by. Generally.
So, from Oxford Dictionary off Google:
Purpose - the reason for which something is done or created or for which something exists.
"the purpose of the meeting is to appoint a trustee"
In the case of animals, we'll be taking that last clause as our meaning. The reason animals exist is their skillsets. It really doesn't have to be any more complicated than that.
We live on a dynamic, ever changing planet. Ome that also holds life that has to maintain itself. That's about more than just procreation. Circle of life and all that.
Life is maintained by the evolutionary engine. The physics of planet create niches where, if exploited by skillsets, life can survive.
Millenia ago, we humans were part of that. But we abstracted ourselves beyond that via civilization and we've been trying to replace that lost sense of belonging, or "purpose", ever since.
Part of that is because civilization means we now exist as a species with a weird duality. We're individuals, but we're also symbiots within civilization, which can be thought of as a system organism.
Civilization is dangerous to us in a way, because it exerts pressure, part of the evolutionary engine. In our case, we're evolving within civilization as either something akin to gut biota or maybe a Volvox colony or were gonna be more like mitochondria or a y other organ cell. Right now, it seems like the mitochondria branch is winning.
I should mention here that as a "Sky Daddy" athiest (that's blunt bit my thoughts on god concepts are not suited for a comment as it's more article worthy) , I don't put much stock in Frankl's philosophic musings, or many philosophical musings for that matter.
That said, he's not wrong about having a purpose helping one endure extreme stress and not give up. But it needs not be a big, grand God given purpose. It could be as simple as telling someone something you want them to know, getting back to loved ones, feeding the cat, etc. Hell, it need not be anything noble or grand either. Look at what Edmond Dantes endured for the sake of revenge in The Count Of Monte Cristo.
Further, I disagree that this phenomenon is solely endowed upon humans. We've all heard stories of lost pets travelling hundreds of miles to return to their families. We see examples of cross species altruism, risking one's own safety to help a trapped individual of another species. Feats of incredible endurance escaping disasters. With the proliferation of filming, we're seeing more and more of all those sorts of things.
What would you call all those things, if not purpose? It's beyond those skillsets you were talking about. Jidt because they can't write sonnets about it that you can understand doesn't mean they're not writing their version of sonnets or art. We just haven't been noticing it any more than we noticed all these incredible fears of purpose before we started putting cameras everywhere and seeing them again.